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Find the number of positive integers n that satisfy \lfloor \sqrt{n} \rfloor = 7 - n.

 Dec 31, 2023
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To find the number of positive integers n that satisfy the equation ⌊√n⌋ = 7 - n, we can follow these steps:

 

Explore Possible Values of ⌊√n⌋:

 

Since ⌊√n⌋ is the greatest integer less than or equal to √n, it can only be a non-negative integer.

 

The right-hand side of the equation, 7 - n, is decreasing as n increases.

 

Therefore, there can only be a finite number of solutions for n.

 

Consider Bounds for n:

 

If n is less than 49, then √n is less than 7, and ⌊√n⌋ will be less than 7 - n.

 

If n is greater than or equal to 49, then √n is greater than or equal to 7, and ⌊√n⌋ will be at least 7.

 

So, we only need to consider values of n between 49 and 4848 (since 48 is the largest integer whose square is less than 487).

 

Check Potential Values of ⌊√n⌋:

 

For n between 49 and 48*48, we can check for values of ⌊√n⌋ that satisfy the equation:

 

If ⌊√n⌋ = 6, then n must be between 36 and 48. There are 13 such values of n.

 

If ⌊√n⌋ = 5, then n must be between 25 and 35. There are 11 such values of n.

 

If ⌊√n⌋ = 4, then n must be between 16 and 24. There are 9 such values of n.

 

If ⌊√n⌋ = 3, then n must be between 9 and 15. There are 7 such values of n.

 

If ⌊√n⌋ is 2 or less, the equation cannot be satisfied for any n between 49 and 48*48.

 

Count Solutions:

 

Adding up the possible values of n for each ⌊√n⌋, we find a total of 13 + 11 + 9 + 7 = 40 possible solutions for n.

 

Therefore, there are 40 positive integers n that satisfy the equation ⌊√n⌋ = 7 - n.

 Dec 31, 2023

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